I recently got back from a summer school in Valladolid on New Perspectives in MCMC.

Here are a few thoughts on the lectures that were given.

Nawaf's interests are mainly in numerical solvers for Stochastic Differential Equations (SDEs) that describe diffusion and drift processes. The first part of the course was about how standard MCMC algorithms can be seen as discretisations of SDEs. The aim of MCMC algorithms is to sample from a probability distribution. A valid MCMC algorithm can be thought of as an SDE discretisation that has the probability distribution of interest as its stationary distribution.

There are also many ways of discretising an SDE that do not correspond to existing MCMC algorithms, and that is what the second part of the course was about. It made me wonder whether there is scope for designing new MCMC methods from these (or other) SDE numerical solvers?

These lectures were about the pseudo-marginal approach, which I have discussed quite a bit in previous blog posts. The key idea is to find ways of approximating the likelihood or the acceptance probability in such a way that resulting algorithm is still a valid MCMC algorithm, but is computationally much cheaper than the simple ideal algorithm (which might, for example, contain an intractable marginal likelihood term in the acceptance ratio).

The thing that I understood better than before was an idea called Rejuvenation. (From what I understand this is the idea used in the particle Gibbs variant of particle MCMC). Without Rejuvenation, the denominator in the acceptance ratio can be very large if the approximation of the likelihood is by chance very poor. This means that the noisy acceptance probability can be much smaller than the acceptance probability of the ideal algorithm, and therefore the noisy algorithm can have a tendency to get stuck. Rejuvenation is a way of revising the noisy estimates in the algorithm from one iteration to the next in a way that preserves the validity of the algorithm and makes it less likely to get stuck.

These lectures had a very clear message that was repeated through-out, which was that thinking in infinite dimensions results in good methodology for high dimensional problems, particularly high dimensional inverse problems and inference for diffusions.

From these lectures I got a much better idea of what it means for measures to be singular with respect to each other. Understanding these concepts is helpful for designing valid proposals for MCMC algorithms in infinite-dimensional spaces, and leads naturally to good proposals in high-dimensional problems. Coincidentally the concepts of singularity and equivalence are also very important in Reversible Jump methodology. For me this illustrates how potent some mathematical ideas are for solving a wide range of problems and for drawing connections between seemingly disparate ideas.

Course website - http://wmatem.eis.uva.es/npmcmc/?pc=41

Here are a few thoughts on the lectures that were given.

__MCMC-based integrators for SDEs (Nawaf Bou-Rabee)__Nawaf's interests are mainly in numerical solvers for Stochastic Differential Equations (SDEs) that describe diffusion and drift processes. The first part of the course was about how standard MCMC algorithms can be seen as discretisations of SDEs. The aim of MCMC algorithms is to sample from a probability distribution. A valid MCMC algorithm can be thought of as an SDE discretisation that has the probability distribution of interest as its stationary distribution.

There are also many ways of discretising an SDE that do not correspond to existing MCMC algorithms, and that is what the second part of the course was about. It made me wonder whether there is scope for designing new MCMC methods from these (or other) SDE numerical solvers?

__Exact approximations of MCMC algorithms (Christophe Andrieu)__These lectures were about the pseudo-marginal approach, which I have discussed quite a bit in previous blog posts. The key idea is to find ways of approximating the likelihood or the acceptance probability in such a way that resulting algorithm is still a valid MCMC algorithm, but is computationally much cheaper than the simple ideal algorithm (which might, for example, contain an intractable marginal likelihood term in the acceptance ratio).

The thing that I understood better than before was an idea called Rejuvenation. (From what I understand this is the idea used in the particle Gibbs variant of particle MCMC). Without Rejuvenation, the denominator in the acceptance ratio can be very large if the approximation of the likelihood is by chance very poor. This means that the noisy acceptance probability can be much smaller than the acceptance probability of the ideal algorithm, and therefore the noisy algorithm can have a tendency to get stuck. Rejuvenation is a way of revising the noisy estimates in the algorithm from one iteration to the next in a way that preserves the validity of the algorithm and makes it less likely to get stuck.

__MCMC in High Dimensions (Andrew Stuart)__These lectures had a very clear message that was repeated through-out, which was that thinking in infinite dimensions results in good methodology for high dimensional problems, particularly high dimensional inverse problems and inference for diffusions.

From these lectures I got a much better idea of what it means for measures to be singular with respect to each other. Understanding these concepts is helpful for designing valid proposals for MCMC algorithms in infinite-dimensional spaces, and leads naturally to good proposals in high-dimensional problems. Coincidentally the concepts of singularity and equivalence are also very important in Reversible Jump methodology. For me this illustrates how potent some mathematical ideas are for solving a wide range of problems and for drawing connections between seemingly disparate ideas.

Course website - http://wmatem.eis.uva.es/npmcmc/?pc=41